Definition

where FF a free abelian group. Since H=ker(F→G)H=ker(F\rightarrow G), it is a free abelian group as well and we choose bases{fα}α\{f_\alpha\}_\alpha for FF and {hβ}β\{h_\beta\}_\beta for HH. We then construct a CW complex

X=M(G,n)
X=M(G,n)

by taking the nn-th skeleton to be Xn:=∨αSαnX^n:=\vee_\alpha S^n_\alpha and for each β\beta we attach an n+1n+1-cell as follows:

Write hβ=Σαdαβfαh_\beta=\Sigma_\alpha d_{\alpha\beta}f_\alpha and let δdαβ\delta_{d_{\alpha\beta}} be 00 if dαβ=0d_{\alpha\beta}= 0 and dαβ=1d_{\alpha\beta}=1 otherwise. Define an attaching map Sβn→XnS^n_\beta\rightarrow X^n by contracting ℓβ:=(Σαδdαβ)−1\ell_\beta:=(\Sigma_\alpha \delta_{d_{\alpha\beta}})-1(n−1)−(n-1)-spheres in SnS^n thus defining a map Sβn→∨ℓβSαβnS^n_\beta\rightarrow \vee_{\ell_\beta} S^n_{\alpha\beta} and then map each SαβnS^n_{\alpha\beta} to SαnS^n_{\alpha} by a degree dαβd_{\alpha\beta}.

Definition

The resulting CW-complex can be seen to have the desired properties via cellular homology.

Properties

Homotopy type

The homotopy type of M(G,n)M(G,n) is determined by specifying GG and nn.

(Non-)Functoriality of the construction

The construction above is not functorial in GG because of the choice of bases (see more below). However, it does give a functor to the homotopy categoryM(−,n):Ab→Ho(Top)M(-,n):Ab\rightarrow Ho(Top).

The functoriality problem of the construction above cannot be corrected. That is, there is no functor Ab→TopAb\rightarrow Top that lifts M(−,n)M(-,n). This can be seen as a corollary of a counterexample of Carlsson which gives a negative answer to a conjecture of Steenrod:

Conjecture

(Steenrod)

Given a group GG a GG-module MM and a natural number nn, there is a GG-space XX which has only one non-zero reduced homology G-module in dimension nn that satisfy H˜n(X;ℤ)≅M\tilde{H}_n(X;\mathbb{Z}) \cong M as GG-modules.

Carlsson provides counter examples for such “equivariant Moore spaces” for all non-cyclic groups.

Corollary

There is thus no functorAb→\rightarrowTop that lifts M(−,n):Ab→Ho(Top)M(-,n)\colon Ab\rightarrow Ho(Top) since if there was such, it would induce, for any group GG a functor AbG→TopGAb^G\rightarrow Top^G and in particular a positive answer to the Steenrod conjecture.

Moreover, there can also not be an (∞,1)-functorAb→LwheTopAb\rightarrow L_{whe} Top that lifts M(−,n)M(-,n) since this will similarly yield an ∞\infty-functor AbG→TophGAb^G\rightarrow Top^{hG} where TophGTop^{hG} is the (∞,1)-category of ∞-actions of GG on spaces. Since there is a “rigidification” functor TophG→TopGTop^{hG}\rightarrow Top^G this would yield an (ordinary) functor AbG→TopGAb^G\rightarrow Top^G which does not exist by our previous observation.

Related concepts

Co-Moore spaces

There is also a cohomology analogue known as a co-Moore space or a Peterson space, but this is not defined for all abelian GG. Spheres are both Moore and co-Moore spaces for G=ℤG = \mathbb{Z}.